In this work we will consider the ringdown phase of exact compact object solutions of scalar-tensor theory, which is part of the Horndeski class of solutions [1]} that give rise to second order field equations in four dimensions [2]}, [3]}, [4]} (for a review of this class of Horndeski theories see [5]}). The BH [6]}-[7]} and wormhole [8]} solutions we consider arise from a gravitational action with a real or phantom scalar field, respectively, non-minimally coupled to the Einstein tensor. These exact solutions encode the `gravitational' scalar, and its coupling strength, in the metric tensor components as a primary charge [9]}, [8]}, which asymptotically plays the role of an effective negative cosmological constant. The motivation for considering these objects is that they have a natural asymptotic reflective boundary, in the external region of the PS, which may lead to different behaviour compared with the case of flat spacetimes. The anti-de Sitter (AdS) spacetime is a necessity for holographic theories which were built by applying the gauge/gravity duality. The aim of holography is to study strongly coupled phenomena using dual gravitational systems where the coupling is weak [11]}. This duality, which is well founded in string theory, has many interesting applications and among them is condensed matter physics. In these theories, AdS BHs play an essential role in the gravity sector in order to achieve the abundant phase structure of the condensed matter system lying on the conformal boundary (for a review, see [12]}). These holographic theories stimulated the extended study of AdS BHs, their formation and their stability, with their QNMs describing the approach to thermal equilibrium in the dual conformal field theory on the boundary [13]}.
